課程資訊
課程名稱
微分幾何一
Differential Geometry (Ⅰ) 
開課學期
106-1 
授課對象
理學院  數學研究所  
授課教師
蔡宜洵 
課號
MATH7301 
課程識別碼
221 U2930 
班次
 
學分
3.0 
全/半年
半年 
必/選修
選修 
上課時間
星期三9(16:30~17:20)星期五3,4(10:20~12:10) 
上課地點
天數102天數102 
備註
總人數上限:80人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1061MATH7301_ 
課程簡介影片
 
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課程概述

This class will go through the following topics in a year.

1. Motivations for manifolds; concept of manifolds; vector fields, tangent vectors, 1-forms
2. Pull-back; push-forward; submanifolds
3. Whitney Embedding theorem; immersions, submersions
4. Tangent bundles; tensors; antisymmetric tensors
5. Flow; one parameter subgroup
6. Brackets, Lie derivative, computation via coordinates
8-10. Differential forms, exterior derivative, tensor operations, contractions;
Interior product, Cartan’s formula, classical vector analysis, Stokes theorem
for differential forms
11-12. Frobenius theorem, Poincare Lemma, De Rham cohomology, differential form version of Frobenius, applications to PDE and geometry
13. Riemannian manifolds, Euclidean case, covariant derivative
14. Riemannian metric, Riemann-Christoffel symbols, parallelism, geodesic
15. Exponential mapping, normal coordinates
16. Convex neighborhoods; two proofs
17. Riemannian curvature, sectional curvature, Ricci curvature
18. Bianchi identity, hypersurface
19. 1st fundamental forms, 2nd fundamental forms determine hypersurfaces up to isometry
20. Variation formula, Gauss lemma
21. Hopf-Rinow theorem
22-24. Jacobi fields, 2nd variation, Jacobi equation, conjugate points, minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs
25-27. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius
28-29. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
30. Cartan-Ambrose-Hicks Theorem
31. Miscellaneous
a) flows and transformations
b) Killing vector fields
c) volume element and divergence
d) Ricci curvature and volume growth
e) 2nd Bianchi identity applied to Einstein manifolds
f) Cut locus, injectivity radius, Klingenberg’s lemma

References for 1st semester:
1. Do Carmo: Riemannian geometry (together with his book on “curves and surfaces”)
2. Gallot-Hulin-Lafontaines: Riemannian geometry
3. Helgason: Differential geometry, Lie groups and symmetric spaces
4.Hicks: Notes on Differential geometry
5. Cheeger-Ebin: Comparison theorems in Riemannian geometry
 

課程目標
待補 
課程要求
待補 
預期每週課後學習時數
 
Office Hours
 
參考書目
待補 
指定閱讀
待補 
評量方式
(僅供參考)
   
課程進度
週次
日期
單元主題